Black Hole Mass/Charge Relation and Weak No-hair Theorem Conjecture

TL;DR

The paper examines four-dimensional Einstein gravity with two Maxwell fields and a real dilaton (EMMD) and analyzes electrically charged, static, spherically symmetric black holes that are asymptotically Minkowski. It derives a universal long-range force relation and homogeneity-based constraints to show that the scalar hair $\Sigma$ is not an independent parameter but a function of $(M,Q_1,Q_2)$, i.e., $\Sigma=\Sigma(M,Q_1,Q_2)$, without solving the full equations of motion. Through exact solutions at special couplings and targeted extremal cases, the authors demonstrate how the mass/charge relations depend on the dilaton couplings $a_1,a_2$ and recover RN limits with $\Sigma=0$, thereby supporting a weak no-hair theorem for a single real scalar. They provide a practical framework to determine $\Sigma(M,Q_1,Q_2)$ from algebraic and differential relations and discuss extensions to theories with more Maxwell fields and dilatons, highlighting implications for horizon data and extremality.

Abstract

We consider Einstein gravity minimally coupled to two Maxwell fields and one (real) dilaton scalar. We study the electrically-charged spherically-symmetric and static solutions that are asymptotic to Minkowski spacetime. General solutions are described by four independent parameters: mass $M$, two electric charges $(Q_1,Q_2)$ and the scalar charge $Σ$. For black holes, the scalar charge is not independent, but a function of $(M,Q_1,Q_2)$. We provide a set of formulae relating the mass to the charges, which allows us to determine $Σ(M,Q_1,Q_2)$ without having to solve the black hole equations. Our results confirm the weaker version of the no-hair theorem conjecture involving one real scalar: it can be turned on in a black hole, but it does not have a continuous independent hairy parameter.

Figures (4)

• Figure 1: In the extremal case $\mu=0$, the function $f(x)$ gives the mass/charge relation, where $M=Q_1 f(x)$ and $x=Q_2/Q_1$. We see that the function is linear if $-a_1 a_2=1$, concave if $-a_1a_2 >1$ and convex when $-a_1a_2<1$.
• Figure 2: Here are the two 3d plots of scalar charge and mass functions $g(x,y)$ and $f(x,y)$ respectively, for the case of $a_1=2$ and $a_2=-1/8$. The flat plane on the bottom in each graph corresponds to the zero value. We see approximately that the scalar charge $g(x,y)=0$ when $y=4 x$, while mass function $f(x,y)$ is always positive.
• Figure 3: In this plot of functions $(f,g)$ with $a_1=2$ and $a_2=-1/8$, we have fixed $x=1/2$. We see that the function $g$ vanishes precisely at $y=2$, where the scalar decouples and the black hole solution becomes the RN black hole.
• Figure 4: For the asymptotic parameters \ref{['m1n1m2']} determined by the differential equations of mass and charges, we find that they indeed give rise to a black hole with both $(H_1,H_2)$ monotonically decreasing from finite values at $r=2\mu$ to 1 at large $r$. The graphs are based on $(a_1,a_2)=(2,-1/8)$. The mass and electric charges are $(M,Q_1,Q_2)=(2.42677,1/2,1)$. Consequently, the scalar charge is $\Sigma=0.166408$.